where \(m:=u-u_{xx}\)  is known as momentum [1]}. The blow up of its solutions, manifested through a singularity in its first order spatial derivative (wave breaking), was proved by Constantin and Escher [2]}, [3]} and [4]} for the non-periodic and periodic cases, respectively. See also [5]}. Moreover, in [6]}, [7]} it was shown that the equation does not have non-zero compactly supported solutions for any \(t\) , except eventually at \(t=0\)  (corresponding to the initial data).
